1123.54/291.51 WORST_CASE(Omega(n^1), ?) 1123.54/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1123.54/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.54/291.52 1123.54/291.52 1123.54/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.54/291.52 1123.54/291.52 (0) CpxTRS 1123.54/291.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1123.54/291.52 (2) TRS for Loop Detection 1123.54/291.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1123.54/291.52 (4) BEST 1123.54/291.52 (5) proven lower bound 1123.54/291.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.54/291.52 (7) BOUNDS(n^1, INF) 1123.54/291.52 (8) TRS for Loop Detection 1123.54/291.52 1123.54/291.52 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (0) 1123.54/291.52 Obligation: 1123.54/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.54/291.52 1123.54/291.52 1123.54/291.52 The TRS R consists of the following rules: 1123.54/291.52 1123.54/291.52 a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) 1123.54/291.52 a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) 1123.54/291.52 a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) 1123.54/291.52 a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) 1123.54/291.52 a__nats(N) -> cons(mark(N), nats(s(N))) 1123.54/291.52 a__zprimes -> a__sieve(a__nats(s(s(0)))) 1123.54/291.52 mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) 1123.54/291.52 mark(sieve(X)) -> a__sieve(mark(X)) 1123.54/291.52 mark(nats(X)) -> a__nats(mark(X)) 1123.54/291.52 mark(zprimes) -> a__zprimes 1123.54/291.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.54/291.52 mark(0) -> 0 1123.54/291.52 mark(s(X)) -> s(mark(X)) 1123.54/291.52 a__filter(X1, X2, X3) -> filter(X1, X2, X3) 1123.54/291.52 a__sieve(X) -> sieve(X) 1123.54/291.52 a__nats(X) -> nats(X) 1123.54/291.52 a__zprimes -> zprimes 1123.54/291.52 1123.54/291.52 S is empty. 1123.54/291.52 Rewrite Strategy: INNERMOST 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1123.54/291.52 Transformed a relative TRS into a decreasing-loop problem. 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (2) 1123.54/291.52 Obligation: 1123.54/291.52 Analyzing the following TRS for decreasing loops: 1123.54/291.52 1123.54/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.54/291.52 1123.54/291.52 1123.54/291.52 The TRS R consists of the following rules: 1123.54/291.52 1123.54/291.52 a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) 1123.54/291.52 a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) 1123.54/291.52 a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) 1123.54/291.52 a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) 1123.54/291.52 a__nats(N) -> cons(mark(N), nats(s(N))) 1123.54/291.52 a__zprimes -> a__sieve(a__nats(s(s(0)))) 1123.54/291.52 mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) 1123.54/291.52 mark(sieve(X)) -> a__sieve(mark(X)) 1123.54/291.52 mark(nats(X)) -> a__nats(mark(X)) 1123.54/291.52 mark(zprimes) -> a__zprimes 1123.54/291.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.54/291.52 mark(0) -> 0 1123.54/291.52 mark(s(X)) -> s(mark(X)) 1123.54/291.52 a__filter(X1, X2, X3) -> filter(X1, X2, X3) 1123.54/291.52 a__sieve(X) -> sieve(X) 1123.54/291.52 a__nats(X) -> nats(X) 1123.54/291.52 a__zprimes -> zprimes 1123.54/291.52 1123.54/291.52 S is empty. 1123.54/291.52 Rewrite Strategy: INNERMOST 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1123.54/291.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1123.54/291.52 1123.54/291.52 The rewrite sequence 1123.54/291.52 1123.54/291.52 mark(nats(X)) ->^+ a__nats(mark(X)) 1123.54/291.52 1123.54/291.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1123.54/291.52 1123.54/291.52 The pumping substitution is [X / nats(X)]. 1123.54/291.52 1123.54/291.52 The result substitution is [ ]. 1123.54/291.52 1123.54/291.52 1123.54/291.52 1123.54/291.52 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (4) 1123.54/291.52 Complex Obligation (BEST) 1123.54/291.52 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (5) 1123.54/291.52 Obligation: 1123.54/291.52 Proved the lower bound n^1 for the following obligation: 1123.54/291.52 1123.54/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.54/291.52 1123.54/291.52 1123.54/291.52 The TRS R consists of the following rules: 1123.54/291.52 1123.54/291.52 a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) 1123.54/291.52 a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) 1123.54/291.52 a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) 1123.54/291.52 a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) 1123.54/291.52 a__nats(N) -> cons(mark(N), nats(s(N))) 1123.54/291.52 a__zprimes -> a__sieve(a__nats(s(s(0)))) 1123.54/291.52 mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) 1123.54/291.52 mark(sieve(X)) -> a__sieve(mark(X)) 1123.54/291.52 mark(nats(X)) -> a__nats(mark(X)) 1123.54/291.52 mark(zprimes) -> a__zprimes 1123.54/291.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.54/291.52 mark(0) -> 0 1123.54/291.52 mark(s(X)) -> s(mark(X)) 1123.54/291.52 a__filter(X1, X2, X3) -> filter(X1, X2, X3) 1123.54/291.52 a__sieve(X) -> sieve(X) 1123.54/291.52 a__nats(X) -> nats(X) 1123.54/291.52 a__zprimes -> zprimes 1123.54/291.52 1123.54/291.52 S is empty. 1123.54/291.52 Rewrite Strategy: INNERMOST 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (6) LowerBoundPropagationProof (FINISHED) 1123.54/291.52 Propagated lower bound. 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (7) 1123.54/291.52 BOUNDS(n^1, INF) 1123.54/291.52 1123.54/291.52 ---------------------------------------- 1123.54/291.52 1123.54/291.52 (8) 1123.54/291.52 Obligation: 1123.54/291.52 Analyzing the following TRS for decreasing loops: 1123.54/291.52 1123.54/291.52 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.54/291.52 1123.54/291.52 1123.54/291.52 The TRS R consists of the following rules: 1123.54/291.52 1123.54/291.52 a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) 1123.54/291.52 a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) 1123.54/291.52 a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) 1123.54/291.52 a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) 1123.54/291.52 a__nats(N) -> cons(mark(N), nats(s(N))) 1123.54/291.52 a__zprimes -> a__sieve(a__nats(s(s(0)))) 1123.54/291.52 mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) 1123.54/291.52 mark(sieve(X)) -> a__sieve(mark(X)) 1123.54/291.52 mark(nats(X)) -> a__nats(mark(X)) 1123.54/291.52 mark(zprimes) -> a__zprimes 1123.54/291.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.54/291.52 mark(0) -> 0 1123.54/291.52 mark(s(X)) -> s(mark(X)) 1123.54/291.52 a__filter(X1, X2, X3) -> filter(X1, X2, X3) 1123.54/291.52 a__sieve(X) -> sieve(X) 1123.54/291.52 a__nats(X) -> nats(X) 1123.54/291.52 a__zprimes -> zprimes 1123.54/291.52 1123.54/291.52 S is empty. 1123.54/291.52 Rewrite Strategy: INNERMOST 1123.86/291.59 EOF